### Working paper

## Noncommutative algebras, context-free grammars and algebraic Hilbert series

The Hilbert series is one of the most important algebraic invariants of innite-dimensional graded associative algebra. The noncommutative Groebner basis machine reduces the prob- lem of nding Hilbert series to the case of monomial algebra. We apply both noncommutative and commutative Groebner bases theory as well as the theory of formal languages to provide a new method for symbolic computation of Hilbert series of graded associative algebras. Whereas in general the problem of computation oh such a Hilbert series is known to be algorithmically unsolvable, we have describe a general class of algebras (called homologically unambiguous) with unambiguous context-free set of relations for which our method give eective algorithms. Unlike previously known methods, our algorithm is applicable to algebras with irrational Hilbert series and produces an alge- braic equation which denes the series. The examples include innitely presented monomials algebras as well as nitely presented algebras with irrational Hilbert series such that the associated monomial algebras are homologically unambiguous.

We consider equilibrium problems for the logarithmic vector potential related to the asymptotics of the HermitePadé approximants. Solutions of such problems can be expressed bymeans of algebraic functions. The goal of this paper is to describe a procedure for determining the algebraic equation for this function in the case when the genus of this algebraic function is equal zero. Using the coefficients of the equation we compute the extremal cuts of the Riemann surfaces. These cuts are attractive sets for the poles of the HermitePadé approximants. We demonstrate the method by an example of the equilibrium problem related to a special system that is called the Angelesco system.

Formal language theory has a deep connection with such areas as static code analysis, graph database querying, formal verifica- tion, and compressed data processing. Many application problems can be formulated in terms of languages intersection. The Bar-Hillel theo- rem states that context-free languages are closed under intersection with a regular set. This theorem has a constructive proof and thus provides a formal justification of correctness of the algorithms for applications mentioned above. Mechanization of the Bar-Hillel theorem, therefore, is both a fundamental result of formal language theory and a basis for the certified implementation of the algorithms for applications. In this work, we present the mechanized proof of the Bar-Hillel theorem in Coq.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

A noncommutative Grassmanian NGr(m,n) is introduced by Efimov, Luntz, and Orlov in `Deformation theory of objects in homotopy and derived categories III: Abelian categories' as a noncommutative algebra associated to an exceptional collection of n-m+1 coherent sheaves on P^n. It is a graded Calabi--Yau Z-algebra of dimension n-m+1. We show that this algebra is coherent provided that the codimension d=n-m of the Grassmanian is two. According to op. cit., this gives a t-structure on the derived category of the coherent sheaves on the noncommutative Grassmanian. The proof is quite different from the recent proofs of the coherence of some graded 3-dimensional Calabi--Yau algebras and is based on properties of a PBW-basis of the algebra NGr(m,n).

It is well known that the composition of a D-finite function with an algebraic function is again D-finite. We give the first estimates for the orders and the degrees of annihilating operators for the compositions. We find that the analysis of removable singularities leads to an order-degree curve which is much more accurate than the order-degree curve obtained from the usual linear algebra reasoning.

The present textbook is intended for students preparing to study mathematics at a higher education institution, to prepare to pass the exam.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.